Document

Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 132, 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)

The Hairy Ball Theorem states that every continuous tangent vector field on an even-dimensional sphere must have a zero. We prove that the associated computational problem of computing an approximate zero is PPAD-complete. We also give a FIXP-hardness result for the general exact computation problem.
In order to show that this problem lies in PPAD, we provide new results on multiple-source variants of End-of-Line, the canonical PPAD-complete problem. In particular, finding an approximate zero of a Hairy Ball vector field on an even-dimensional sphere reduces to a 2-source End-of-Line problem. If the domain is changed to be the torus of genus g >= 2 instead (where the Hairy Ball Theorem also holds), then the problem reduces to a 2(g-1)-source End-of-Line problem.
These multiple-source End-of-Line results are of independent interest and provide new tools for showing membership in PPAD. In particular, we use them to provide the first full proof of PPAD-completeness for the Imbalance problem defined by Beame et al. in 1998.

Paul W. Goldberg and Alexandros Hollender. The Hairy Ball Problem is PPAD-Complete. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 65:1-65:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{goldberg_et_al:LIPIcs.ICALP.2019.65, author = {Goldberg, Paul W. and Hollender, Alexandros}, title = {{The Hairy Ball Problem is PPAD-Complete}}, booktitle = {46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)}, pages = {65:1--65:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-109-2}, ISSN = {1868-8969}, year = {2019}, volume = {132}, editor = {Baier, Christel and Chatzigiannakis, Ioannis and Flocchini, Paola and Leonardi, Stefano}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2019.65}, URN = {urn:nbn:de:0030-drops-106416}, doi = {10.4230/LIPIcs.ICALP.2019.65}, annote = {Keywords: Computational Complexity, TFNP, PPAD, End-of-Line} }

Document

**Published in:** LIPIcs, Volume 117, 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018)

The Consensus-halving problem is the problem of dividing an object into two portions, such that each of n agents has equal valuation for the two portions. We study the epsilon-approximate version, which allows each agent to have an epsilon discrepancy on the values of the portions. It was recently proven in [Filos-Ratsikas and Goldberg, 2018] that the problem of computing an epsilon-approximate Consensus-halving solution (for n agents and n cuts) is PPA-complete when epsilon is inverse-exponential. In this paper, we prove that when epsilon is constant, the problem is PPAD-hard and the problem remains PPAD-hard when we allow a constant number of additional cuts. Additionally, we prove that deciding whether a solution with n-1 cuts exists for the problem is NP-hard.

Aris Filos-Ratsikas, Søren Kristoffer Stiil Frederiksen, Paul W. Goldberg, and Jie Zhang. Hardness Results for Consensus-Halving. In 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 117, pp. 24:1-24:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{filosratsikas_et_al:LIPIcs.MFCS.2018.24, author = {Filos-Ratsikas, Aris and Frederiksen, S{\o}ren Kristoffer Stiil and Goldberg, Paul W. and Zhang, Jie}, title = {{Hardness Results for Consensus-Halving}}, booktitle = {43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018)}, pages = {24:1--24:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-086-6}, ISSN = {1868-8969}, year = {2018}, volume = {117}, editor = {Potapov, Igor and Spirakis, Paul and Worrell, James}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2018.24}, URN = {urn:nbn:de:0030-drops-96069}, doi = {10.4230/LIPIcs.MFCS.2018.24}, annote = {Keywords: PPAD, PPA, consensus halving, generalized-circuit, reduction} }

Document

**Published in:** LIPIcs, Volume 94, 9th Innovations in Theoretical Computer Science Conference (ITCS 2018)

The class TFNP, of NP search problems where all instances have solutions, appears not to have complete problems. However, TFNP contains various syntactic subclasses and important problems. We introduce a syntactic class of problems that contains these known subclasses, for the purpose of understanding and classifying TFNP problems. This class is defined in terms of the search for an error in a concisely-represented formal proof. Finally, the known complexity subclasses are based on existence theorems that hold for finite structures; from Herbrand's Theorem, we note that such theorems must apply specifically to finite structures, and not infinite ones.

Paul W. Goldberg and Christos H. Papadimitriou. Towards a Unified Complexity Theory of Total Functions. In 9th Innovations in Theoretical Computer Science Conference (ITCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 94, pp. 37:1-37:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{goldberg_et_al:LIPIcs.ITCS.2018.37, author = {Goldberg, Paul W. and Papadimitriou, Christos H.}, title = {{Towards a Unified Complexity Theory of Total Functions}}, booktitle = {9th Innovations in Theoretical Computer Science Conference (ITCS 2018)}, pages = {37:1--37:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-060-6}, ISSN = {1868-8969}, year = {2018}, volume = {94}, editor = {Karlin, Anna R.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2018.37}, URN = {urn:nbn:de:0030-drops-83403}, doi = {10.4230/LIPIcs.ITCS.2018.37}, annote = {Keywords: Computational complexity, first-order logic, proof system, NP search functions, TFNP} }

Document

**Published in:** Dagstuhl Reports, Volume 7, Issue 6 (2018)

his report documents the program and the outcomes of Dagstuhl Seminar 17251
"Game Theory Meets Computational Learning Theory".
While there have been many Dagstuhl seminars on various aspects of Algorithmic
Game Theory, this was the first one to focus on the emerging field of
its intersection with computational learning theory.

Paul W. Goldberg, Yishay Mansour, and Paul Dütting. Game Theory Meets Computational Learning Theory (Dagstuhl Seminar 17251). In Dagstuhl Reports, Volume 7, Issue 6, pp. 68-85, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@Article{goldberg_et_al:DagRep.7.6.68, author = {Goldberg, Paul W. and Mansour, Yishay and D\"{u}tting, Paul}, title = {{Game Theory Meets Computational Learning Theory (Dagstuhl Seminar 17251)}}, pages = {68--85}, journal = {Dagstuhl Reports}, ISSN = {2192-5283}, year = {2017}, volume = {7}, number = {6}, editor = {Goldberg, Paul W. and Mansour, Yishay and D\"{u}tting, Paul}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagRep.7.6.68}, URN = {urn:nbn:de:0030-drops-82876}, doi = {10.4230/DagRep.7.6.68}, annote = {Keywords: Algorithmic Game Theory, Computational Learning Theory, Economics} }

Document

Tutorial

**Published in:** LIPIcs, Volume 30, 32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015)

Game theory studies mathematical models of interactions amongst self-interested entities. A "solution concept" means a description of the outcome of a game, and it is important that it should be defined in such a way that a solution always exists (every game should have an outcome). Nash's famous theorem that mixed-strategy equilibria are guaranteed to exist, resulted in Nash equilibrium being the most prominent solution concept in game theory.
As a result, computational challenges of the form "given a game, find a solution", have the property that we are searching for something whose existence is guaranteed (they are total search problems). Moreover, these solutions belong to the complexity class NP, since it is usually straightforward to check whether a proposed solution is correct (an incorrect one will admit a profitable deviation by one or more of the players, and this is usually easy to find). However, in versions of the problem that appear to be computationally hard, we cannot apply NP-completeness, due to a result of Megiddo saying that total search problems cannot be NP-complete unless NP is equal to co-NP.
In this tutorial, which is intended for people familiar with NP-completeness, I give an overview of the alternative notions of computational hardness that apply to game-theoretic solution concepts. I discuss the complexity class PPAD (introduced by Papadimitriou) which captures the computational complexity of various classes of games that don’t seem to be solvable in polynomial time. I also mention the complexity classes PLS and FIXP, and the kinds of games that they apply to.
Suppose, alternatively, that we have a polynomial-time algorithm that applies to some given class of games. A follow-up question is whether there exist algorithms that find a solution via processes that reflect decentralised selfish behaviour. This is because a solution concept arguably remains unrealistic if it can be efficiently computed, but only using a highly centralised algorithm. In the second half of the tutorial I present some results on learning dynamics for equilibrium computation, and mention recent work on communication complexity and query complexity.
I discuss some research directions and open problems, such as the following. What are the prospects for proving that PPAD is as hard as NP? How about algorithms that find improved approximate Nash equilibria? 2-player games are easy to solve in practice, using the Lemke-Howson algorithm, so is there a satisfying mathematical sense in which 2-player games are easy to solve? (For example, a sense in which Lemke-Howson works "most of the time"?)

Paul W. Goldberg. Algorithmic Game Theory (Tutorial). In 32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 30, p. 20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{goldberg:LIPIcs.STACS.2015.20, author = {Goldberg, Paul W.}, title = {{Algorithmic Game Theory}}, booktitle = {32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015)}, pages = {20--20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-78-1}, ISSN = {1868-8969}, year = {2015}, volume = {30}, editor = {Mayr, Ernst W. and Ollinger, Nicolas}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2015.20}, URN = {urn:nbn:de:0030-drops-49606}, doi = {10.4230/LIPIcs.STACS.2015.20}, annote = {Keywords: equilibrium, non-cooperative games, computational complexity} }

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